The recent generalizations

of colored symmetry

Lungu Alexandru

Department of Mathematics and InformaticsState University of Moldovastr. A. Mateevici 60MD 2009 ChisinauMoldovalungu@usm.md

Abstract. In this article we analyze the
essence of the recent "physical'' generalizations of classical symmetry
from the viewpoint of the split extensions of groups. The principal results
of the general theory of groups of Wp- and Wq-symmetry
are given. The methods of deriving generalized symmetry groups of different
types are formulated. All these methods are based on right, left and crossed
quasi-homomorphisms and their generalizations. The principal properties
of diverse quasi-homomorphic mappings are examined. The article is illustrated
with many examples of colored and ïndexed" geometrical figures that
are described by the groups of P-,`P-,
Wp-
or Wq-symmetry.

It is well known that the modern theory of symmetry of
the real crystal gives rise to new generalizations.

One of the essential generalizations of classical symmetry
is the P-symmetry of A.M.Zamorzaev [1-3]. In
the case of P-symmetry, the transformations of the qualities attributed
to the points, are combined directly with the geometrical transformations
and do not depend on the choice of points. Other proposed generalizations
such as polychromatic symmetry of Wittke-Garrido [4]
or complex [5] symmetry are not included in the scheme
of A.M.Zamorzaev's P-symmetry. For these generalizations, the transformations
of the qualities attributed to the points, essentially depend on the choice
of points. The mentioned generalizations are included in Wp-symmetry
that was introduced by V.A.Koptsik and I.N.Kotsev [6]
and was developed in the cycle of papers [3, 6-21,
28,
33,
35].

On the other hand, V.A.Koptsik in [22]
proposed the notion of Q-symmetry. The essence of Q-symmetry
consists actually in the following: the transformation of Q-symmetry
g(q)
is composed from the components g and q, where g is
transformation of symmetry which operates both on points (the atoms of
crystal) and on indexes (vector or tensor) by the given rule independent
of the points, and q is a supplementary transformation of these
indexes. Uniting problems to be solved for Wp-symmetry
and Q-symmetry (`P-symmetry [23-25]),
we shall get a generalization named Wq-symmetry [26-33].

The methods of deriving the groups of P-symmetry
of different types [1, 2] are based
on homomorphic mapping and its properties. The solution of analogous problems
for`P]-
and W-symmetry demands the generalization of homomorphisms as the
right quasi-homomorphism [23, 3],
the natural left quasi-homomorphism [14], and the
crossed quasi-homomorphism [27]. Moreover, it requires
the investigation of some their properties.

2. The crossed standard Cartesian

wreath product of groups

Let us have groups G and P. Construct the
Cartesian product W of isomorphic copies of the group P which
are indexed by elements of G: W =`ÕgiÎ
GPgi,
where Pgi
@P.
Moreover, construct the isomorphic injection f:
G®Aut
W by the rule f(g) =
(where the automorphism
makes the left g-translation of the components in wÎW,
i.e. :
w®wg),
and also the homomorphism
t: G®
F £Aut W (where
t(g)
= and (w)
= gwg-1). The set G* of pairs wg
(where wÎW and gÎG)
forms a group with the operation
wigi*
wjgj
= wkgk, where gk = gigj
and wk = wigj(wj).
We call G* the crossed standard Cartesian wreath
product of groups P and G, accompanied with the homomorphism
t
: G®Aut W by the rule t(gi)
= , and
denote it by the symbol P
G`[26,
28,
31].

If W is a direct product of the isomorphic copies
of the group P which are indexed by elements of G, then at
analogous mode it defined the crossed standard direct wreath product
of groups P and G.

3. The essence of the recent generalizations

of color symmetry

Ascribe to each point of a geometrical figure F
with the discrete symmetry group G at least one index, which means
a non-geometrical feature, from the set N = {1,2,...,m},
and fix a certain transitive group P of the permutations of these
indexes.

The transformation of P-symmetry is defined to
be an isometric mapping g(p) = gp = pg
of the "indexed" geometrical figure F(N) onto
itself in which the geometrical component g operates only on points,
and the indexes are transformed by the permutation p of the group
P.
The set G(P) of transformations of P-symmetry
of any "indexed'' geometrical figure F(N) forms
a group with the operation

gi(pi)*gj(pj)
= gk(pk),
(1)

where gk = gi gj
and pk = pi pj [1].
The groups G(P) of P-symmetry are subgroups
of the direct products of the group P of permutations with their
generating discrete groups G of classical symmetry.

with the group/subgroup symbol 4/1, where 4 is the symbol
of the generating group and 1 is the symbol of its classical symmetry subgroup
[2].

The transformation of `P-symmetry
is defined to be an isometric mapping g(p) = pg
of the "indexed'' geometrical figure F(N) onto
itself in which the geometrical component g operates both on points
and on indexes by the given rule independent of the points, but the permutation
p
is only a compensating permutation of indexes to map F(N)
onto itself and pÎP. In
this case the components p and g of the transformation g(p)
in general do not commute: pg¹gp,
that is why p¹gpg-1.

The set G(`P)
of transformations of `P-symmetry of
any "indexed'' geometrical figure F(N) forms a
group with the operation

gi(pi)*gj(pj)
= gk(pk),
(2)

where gk = gi gj,
pk
= pi(pj)
and (pj)
= gi pj gi-1 = psÎP.
The groups of `P-symmetry are subgroups
of the right semi-direct product of the group P with group
G,
accompanied with the homomorphism f:
G®Aut
P by the rule f(gi) =
[23,
3].

Example 2. Let us have six different positions
of vectors with the equal modules, that are numbered using 1,2,3,4,5 and
6 in accordance with the scheme represented in Figure 2.

Figure 2

In this case, the "ïndexed" figures represented in
Figure 3a) and Figure 3b) are described from the viewpoint of P-symmetry
by the same group with the group/subgroup symbol 4/1.

Figure 3

From the viewpoint of `P-symmetry
the "ïndexed" figures represented in Figures 3a) and 3b) are described
by the different groups [3].

Construct the Cartesian product W of isomorphic
copies of the group P which are indexed by elements of G,
i.e. W = `ÕgiÎ
GPgi,
where Pgi
@P.The transformation of Wp-symmetry is
defined to be an isometric mapping g(w) = gw
of the "ïndexed" geometrical figure F(N)
onto itself in which the geometrical component g operates only on
points Mk = gk(M1)
of the figure F(N) (where M1
is a point of general position of the figure F with respect to the
group G), not affecting indexes, and the indexes ascribed to the
point Mk are transformed by the permutation pgk
which is the "gk-component'' in w. The set of
transformations of Wp-symmetry of the given "indexed''
geometrical figure F(N) forms a group G(Wp)
with the operation

Example 3. Let's have the groups G = 4 and
P
= (e, p = (12)), where the yellow and red colors are denoted
with indexes 1 and 2. Then g(w) =
gw =
4 <p1,
e4,
p2,
e4-1> is a transformation of
Wp-symmetry
of the colored square represented in Figure 4.

Figure 4

We note that in the case when G = 4 and P =
(e, p = (12)), the transformation of Wp-symmetry
g(w)
= 4 < p1,
p4,
p2,
p4-1> exists too, which may be formally considered as a transformation
of P-symmetry (it satisfies the conditions of the respective definition
- there is one rule of the transformation of colors for all equivalent
points) [35]. This transformation and the transformation
of P-symmetry g(p) = gp = 4p
describe the same colored square (Figure 5).

Figure 5

The groups G(Wp) of
Wp-symmetry
are subgroups of the left standard Cartesian wreath product of the initial
group P of permutations with the discrete group G of classical
symmetry as their generating group:

G(Wp)£G

P = GW.

The transformation of Wq-symmetry is
defined to be an isometric mapping g(w) = wg
of the "indexed'' geometrical figure F(N) onto
itself in which the geometrical component g operates both on points
Mk
= gk(M1) of the figure F(N)
(where M1 is a point of general position of the figure
F
with respect to the group G) and on indexes by the given rule independent
of the points, but the permutation
pgk ("gk-component''
in
w) is only a compensating permutation of indexes in the point
Mk
to map F(N) onto itself. The set G(Wq)
of transformations of
Wq-symmetry of the given "indexed''
figure F(N) forms a group with the operation

The groups G(Wq) of
Wq-symmetry
are subgroups of the crossed standard Cartesian wreath product of the initial
group P of permutations and the discrete group G of classical
symmetry (as their generating group), accompanied with the homomorphism
t:
G®Aut
W by the rule t(gi) =:

G(Wq)£P

G = W

G.

4. On the classification and the general structure

of W-symmetry groups

Let G(W) be a group of Wp-
or
Wq-symmetry with the initial group P, generating
group G and subset W¢
= {w | g(w)ÎG(W)}
ÍW.
Identifying the groups G and W with their isomorphic injections
into GW
= G P
(into
WG
= PG,
respectively) by the rules g®gw0,
where w0 is the unit of the group
W, and w
1w, where 1 is the unit of the group G (g w0g
and ww1,
respectively), we find the symmetry subgroup H = G(W)Ç
G
and the subgroup V = G(W)Ç
W
= G(W)Ç
W¢
of W-identical transformations of the group G(W).

The group G(W) is called senior,
junior or V-middle if w0 <
V
= W¢ = W, w0
= V < W¢
= W or w0 < V
< W¢ = W,
respectively. If W¢ is
a non-trivial subgroup of W, then the group G(W)
is called W¢-semi-senior,
W¢-semi-junior
or (W¢,
V)-semi-middle
according to the cases when w0 <
V
= W¢,
w0
= V < W¢
or w0 < V
< W¢. If W¢ÌW,
but W¢ is not a group,
the group G(W) is called W¢-pseudo-junior
or (W¢, V)-pseudo-middle
when w0 = VÌW¢
or w0 < VÌW¢.

Let G(Wp) be a group
of Wp-symmetry with the initial group P, generating
group G, subset W¢
= {w| g(w)ÎG(Wp)},
symmetry subgroup H and the subgroup V of W-identical
transformations. Then:

1) the mapping
f
of the group G(Wp) onto the group G
by the rule f[g(w)]
= g is homomorphic with the kernel V;

2) the group
G(Wp)
contains as its subgroup the group G1(W1)
of P-symmetry (which is determined by initial group P of
permutations, where W1£Diag
W@P and W1ÌW¢)
from the family with the generating group G1 (G1
£
G),
with the symmetry subgroup
H (where HG1
but H¹ G) and with the subgroup
V1
of W-identical transformations (where V1 = V
Ç
Diag
W).

Moreover, if W is a finite group then:

1) Vg
= wVw-1, where g and w are components of
the transformation g(w) from G(Wp);

2) all the
elements of a right coset Hg of the group G by H are
combined in pairs only with the elements of one left coset wV, and
the elements of different cosets Hgi and Hgj
with the elements of different cosets wiV and wjV
[18-20, 28].

Let G(Wq) be a group
of Wq-symmetry with the generating group G, permutation
group P, (i.e. W = `ÕgiÎ
GPgi,
where Pgi
@P),
subset W¢ = {w |
g(w)
Î
G(Wq)},
with the kernel H1 of accompanying homomorphism t:
G
®
Aut
W, symmetry subgroup
H and the subgroup V of W-identical
transformations. In this case the following conditions are satisfied:

1) the mapping
f
of the group G(Wq) onto the generating
group G by the rule f[g(w)]
= g is homomorphic with the kernel V;

2) the group
G(Wq)
contains as subgroup the group
H1(Wp)
of Wp-symmetry (which is determined by the initial group
P
of permutations) from the family with the generating group H1,
with the symmetry subgroup H¢
(where H¢ = H Ç
H1)
and with the same subgroup V of W-identical transformations;

3) the group
G(Wq)
contains as a subgroup the group
G1(W1)
of `P-symmetry (which is determined by
initial group P of permutations) from the family with the generating
group G1 (where G1£G),
with the same kernel H1 of the accompanying homomorphism
and with the set W1 = {w|
g(w)
Î
G1(W1)}
of multi-component permutations, where W1 = W¢
Ç
Diag
W.

Let us have groups G and P and a homomorphism
f:
G®Aut
P. The mapping y of the group G onto
the subset P¢ of the
group P by the rule y(g) =
p
is called a right quasi-homomorphism if for any
gi
and gjfrom G

y(gigj)
= y(gi)

[y(gj)]
= pi

(pj) = pk,

where pi, pj, pkÎP¢
and
= f(gi)
[3]. The mapping f is called
the accompanying homomorphism of right conjugation. Under the same initial
conditions the mapping
m is called a left
quasi-homomorphism, accompanied by the homomorphism
fof
left conjugation, if

m(gigj)
= [m(gi)]

m(gj)
= (pi)

pj
= pk.

When Kerf=1, the mapping
m
is called a natural left quasi-homomorphism [14].

At the right quasi-homomorphism y,
in general, the image of Gy(G)
= P¢ÌP
is not a group, but P¢
always contains the unit of the group P. The kernel H of
the right quasi-homomorphism
y of the group
G
into the group P is a subgroup in G; the index of this subgroup
coincides with the power of y(G).

The mapping
of the group G onto the subset X of the set of all the right
cosets of group P by its subgroup Q (Q
< P) is called a generalized right quasi-homomorphism
if for any gi and gj from G
conditions (gi)
= Qpi and (gj)
= Qpj imply

The necessary and sufficient condition for the mapping
of the group G onto the subset
X of the set of all the right
cosets of group P by its subgroup
Q by the rule (g)
= Qp to be a generalized right quasi-homomorphism is
(Q)
= p-1Qp for any gÎG
and Qp = (g)
[3].

6. On the methods for deriving groups

of P- and `P-symmetry

Any group G(P) of complete P-symmetry
(P¢ = P) can be
derived from its generating group G and permutation group P
by the following steps:

1) to find
in G and in P all invariant subgroups H and Q
for which there is the isomorphism of factor-groups S/H and
P/Q
(l: G/H®
P/Q)
by the rule l(gH) = pQ;

2) to combine
pair-wise each g¢ of
gH
with each p¢ of pQ
=
l(gH);

3) to introduce
into the set of all these pairs the operation (1).

If Q = e (G(P)
is a junior group), then the isomorphism l:
G/H
®
P
is, in fact, a homomorphism of the group G onto
P (i.e. it
is a representation of the group
G) with the kernel
H [1,
2].

Example 5. In the case when the generating group
G
= {a, b} (4) (we use Zamorzaev's symbols [2])
and the permutation group P = (e, p1 =
(1234), p2 = (13)(24), p3 = (1432))
@
4, we have only three different junior groups of 4-symmetry. Their geometrical
interpretations are represented by colored periodical mosaics (Figures
9 - 11).

Any group of `P-symmetry
(with a finite group P) can be derived from its generating group
G,
knowing the kernel H of accompanying homomorphism f:
G
®
Aut
P, by the following steps:

1) to find
in P all subgroups Q and subsets P¢,
which are decomposed in right cosets by its subgroup Q, and in G
all proper subgroups H¢
(H¢<
G) with
the index equal to the power of set of all the right cosets of P¢by
Q and for which there is the isomorphism l
of factor-groups H/H¢¢
and P¢¢/Q (l:
H/H¢¢
® P¢¢/Q)
by the rule l(gH¢¢)
= pQ) where eQP¢¢ÍP¢ÍP,
P¢¢
< P and H¢¢
= H¢Ç
HH;

2) to construct
a generalized right quasi-homomorphism
of the group G onto the set of all the right cosets of P¢
by the subgroup Q by the rule (gH¢)
= Qp and with accompanying homomorphism f:
G
®
Aut
P with the kernel
H;

3) to combine
pair-wise each g¢ of
gH¢
with each p¢ of Qp
=
(g¢);

4) to introduce
into the set of all these pairs the operation (2).

If Q = e, then the mapping
is an ordinary right quasi-homomorphism and the universal method of deriving
the groups of `P-symmetry becomes more
simple and takes the form of the method of deriving the junior, semi-junior
or pseudo-junior groups in dependence on P¢
[3].

7. On the generalized exact natural left

quasi-homomorphisms and the groups of Wp-symmetry

The natural left quasi-homomorphism m
of the group G into the group W = `ÕgiÎ
GPgi
under which the automorphism
operates on the elements
w of
m(G)
by means of the left
g-translations of their components is called
an exact natural left quasi-homomorphism.

The necessary and sufficient condition for the mapping
m
of the group G onto the subset W¢
of the group W = `ÕgiÎ
GPgi
by the rule m(g) = w to be an
exact natural left quasi-homomorphism is that (wi)wj
= wigjwj = wkÎW¢
for any gjÎG
and wj =
m(gj).
Moreover, the necessary and sufficient condition for the exact natural
left quasi-homomorphism
m of the group G
onto the subgroup W¢
of W to be an ordinary homomorphism is W¢
£
Diag
W. We note that in the case of the exact natural left quasi-homomorphism
m
of the group
G onto the subgroup W¢
of the group W with the kernel Kerm
= H the followings conditions are not compatible:

Let us have the group G, the finite group W
= ÕgiÎ
GPgi,
its subgroup V (V < W)
and the exact isomorphic injection f of the
group G into the subgroup
of the group Aut W by the rule f(g)
= . The
mapping
of
the group G onto the subset
X of the set of all left cosets
of group W by its subgroup
V is called a generalized
exact natural left quasi-homomorphism if for any gi
and gj from G conditions (gi)
= wiV and (gj)
= wjV imply

(gigj)
= (wiV)gj*wjV
= wkV,

where wiV, wjV, wkVÎX.
Note that in the case of V = w0 the mapping
is an ordinary exact natural left quasi-homomorphism.

The necessary and sufficient condition for the mapping
of the group G onto the subset
X of the set of all left cosets
of the finite group W = ÕgiÎ
GPgi
by its subgroup V by the rule
(g) = wV to be a generalized exact natural left quasi-homomorphism
is Vg = wVw-1 for any gÎG
and wV = (g).

If VW, then the generalized exact natural left quasi-homomorphism
of the group G onto the subset X of the set of all left cosets
of group W by its subgroup V is an ordinary left quasi-homomorphism
m
of the group G onto the subgroup X of the factor-group W/V.
In this case the mapping
is accompanied by homomorphism :
G ®
Aut W/V with the
kernel
H, where H@ F0<
Aut W and F0 is the
set of the automorphisms
of group W (and Î Aut W) that generates the
identical automorphism of factor-group
W/V [36].

Any group G(Wp) of
Wp-symmetry
with the finite group W can be derived from its finite generating
group G and a group W = ÕgiÎ
GPgi
of multi-component permutations by the following steps:

1) to find
in W all subgroups V and subsets W¢,
which are decomposed in left cosets by its subgroup V, and in G
all proper subgroups H with the index equal to the power of set
of all left cosets of W¢
by V and for which there is the isomorphism l
of factor-groups G1/H and W1/V1
(l: G1/H®
W1/V
by the rule l(Hg) = wV), where
G1£
G,
W1
£Diag
W and
V1 = VÇ
Diag
W£
W1;

2) to construct
a generalized exact natural left quasi-homomorphism
of the group G onto the set of all left cosets of W¢
by the subgroup V by the rule (Hg)
= wV and which preserves the correspondence between the elements
of factor-groups G1/H and W1/V1
obtained as the result of isomorphism l;

3) to combine
pair-wise each g¢ of
Hg
with each w¢ of wV
= (g¢);

4) to introduce
into the set of all these pairs the operation (3) [16-21,
28].

If V = w0, where w0
is the unit of the group W, then the mapping
is an ordinary exact natural left quasi-homomorphism. In this case, the
universal method of deriving the groups of Wp-symmetry
becomes more simple and takes the form of method for deriving the semi-junior
or pseudo-junior groups in dependence on W¢,
where W¢ÌW
[14].

For the groups G(Wp)
of Wp-symmetry the polynomial symbol (the symbol is formed
from more terms) was proposed in the form

G/(P|
W¢|
V|
W1
/ V1;
G1/H¢/H),

where

1) G
is the generating group for G(Wp);

2) P
is the initial group of permutations;

3) W¢
= {w| g(w)Î
G(Wp)}
Í
W;

4) V
is the subgroup of W-identical transformations;

5) H
is the classical symmetry subgroup;

6) G1/H¢/H
is the trinomial symbol (the three-term symbol) for the group of P-symmetry
G1(W1)
in accordance with [34, 2];

7) W1
/ V1@G1/H,
where V1 = VÇ Diag
W£W1 and W1 = W¢Ç
Diag
W.

In the case of semi-junior and pseudo-junior groups the
polynomial symbol becomes simpler and takes the form

We note that the colored square represented in Figure
5 is described by the semi-junior group of Wp-symmetry
G1(Wp)
= (1w0, 4w, 2w0, 4-1w),
where w = <
p1,
p4,
p2,
p4-1> ; the polynomial abbreviated symbol of this group has the form
4/(4/2/2). Moreover, the same colored square is described by the junior
group of P-symmetry
G(P) = (1e,
4p, 2e, 4-1p), with the three-term symbol
4/2/2. The group G1(Wp)
may be formally considered as a group of P-symmetry.

The square with classical symmetry group G = C4
may be painted with two colors by one more mode [35]
(Figure 12).

Figure 12: Geometrical interpretation of the pseudo-junior
group of Wp-symmetry with the polynomial
abbreviated symbol 4/(1/1/1)

8. The crossed quasi-homomorphisms

and the semi-junior groups of Wq-symmetry

Let us have groups G, P and W = `ÕgiÎ
GPgi
(where Pgi
@P),
the isomorphic injection f:
G®
Aut
W by the rule f(g) = ,
where the automorphism
makes the left g-translation of the components in wÎW (i.e. :
w®wg),
and also the homomorphism
t:
G®
F £Aut W (where
t(g)
= and
(w)
= gwg-1). The mapping
a of
the group G onto the subset W¢
of the group
W by the rule a(g)
= w is called a crossed quasi-homomorphism, accompanied
by the exact left translation of components and by the homomorphism t
of right conjugation, if for any gi and gj
from G

a(gigj)
= [a(gi)]gj*

[a(gj)]
= wigj

(wj) = wk,

where wi, wj, wkÎW¢.

We note that in the case of
= i (where i is the identical automorphism of group W
for any g from G) the crossed quasi-homomorphism a
is an ordinary exact natural left quasi-homomorphism; if wg
= w for all gÎG
and wÎa(G),
then a is right quasi-homomorphism accompanied
by the homomorphism t of right conjugation.

In general, the image of G,a(G)
= W¢ÌW
is not a group, but W¢
always contains the unit of the group W. The kernel H of
crossed quasi-homomorphism
a of the group G
in the group W is a subgroup of the group G.

Let us have crossed quasi-homomorphism a
with Kera = H of the group G
onto the subset W¢ of
the group W accompanied by exact left translations of components
and by the homomorphism t of right conjugation
with Kert = H1. Then:

1) the necessary
and sufficient condition for the mapping
a to
send each left coset gH onto the only element w is that wh
= w for any hÎH
= Kera and wÎW¢;

2) if Ker
a
£Ker t, then a
sends each right coset Hg of group
G by its subgroup H
= Ker a onto the only element wÎ
W¢.

Moreover, if the kernel Kera
= H of the mapping
a is an invariant
of the group G and H£Kert,
then:

a) the crossed
quasi-homomorphism
a sends each left coset gH
onto the only element wÎW¢; b) the equalities
wh
= w are valid for any hÎH
and for any w from W¢;

Any semi-junior group G(Wq)
of Wq-symmetry can be derived from its generating group
G
and group W = `ÕgiÎ
GPgi,
knowing the kernel
H1 of accompanying the homomorphism
t:
G
®
Aut
W of right conjugation, by the following steps:

1) to construct
a crossed quasi-homomorphism a of the group
G
onto the non-trivial subgroup W¢
of W by the rule a(g) = w;

2) to combine
pair-wise each g of G with each w = a(g);

3) to introduce
into the set of all these pairs the operation (4) [27-29].

with P@C4
and w =
<
p21,
p24,
p22,
p24-1,
p2m1,
p2m2,
p2m3,
p2m4
> . Moreover, the same "ïndexed" figure is described by the
semi-junior group of `P-symmetry

G2(`P)
= (e1, e2, em2, em4,
p24,
p24-1,
p2m1,
p2m3).

The group G1(Wq)
may be formally considered as a group of `P-symmetry.

9. The generalized crossed quasi-homomorphisms

and the middle groups of Wq-symmetry

Let us have the group G, the finite group W
= ÕgiÎ
GPgi
(where Pgi
@P),
its subgroup V, the isomorphic injection f:
G®
Aut
W by the rule f(g) =
(where
: w®wg) and also the homomorphism
t: G®
Aut
W (where t(g) =
and
(w)
= gwg-1). The mapping
of the group G onto the subset X of the set of all right
cosets of group W by its subgroup V is called a generalized
crossed quasi-homomorphism, accompanied by exact left translation
of components and by homomorphism
t of right
conjugation, if for any gi and gj from
G from the conditions (gi)
= Vwi and (gj)
= Vwj it follows that

(gigj)
= (Vwi)gj*

(Vwj) = Vwk,

where Vwi, Vwj, VwkÎX
and =
t(gi)
Î
F £Aut W.

We note that in the case of V = w0
the generalized crossed quasi-homomorphism
is an ordinary crossed quasi-homomorphism. If tg
= i for any g from G, then the mapping
is a generalized exact natural left quasi-homomorphism. Moreover, if wg
= w for any g from G and w from W¢=
(G),
then is a generalized right quasi-homomorphism
accompanied by the homomorphism
t of right conjugation.

Any middle group G(Wq)
of Wq-symmetry with the finite group W and the
subgroup V of W-identical transformations can be derived
from its finite generating group G and the group W = ÕgiÎGPgi
of multi-component permutations by the following steps:

1) to find
in W all proper -invariant
subgroups V (w0<
V<
W);

2) to construct
a generalized crossed quasi-homomorphism
of the group G onto the set of all right cosets of the group W
by the subgroup V by the rule (g)
= Vw;

3) to combine
pair-wise each g of G with each w¢
of Vw = (g);

4) to introduce
into the set of all these pairs the operation (4) [28-33].

Remark that in operation (4) there are two different automorphisms
and , which
are independent and, therefore, by consecutive action on the same
w
they commute.

[33] Lungu A., The discrete groups of generalized
symmetry and the quasi-homomorphic mappings. Scientific Annals Faculty
of Mathematics and Informatics. State University of Moldova, Chisinau,
1999, p. 115-124.

[35] Lungu A.P., On coloring figures by different
colors and their description in the framework of generalized color symmetry.
Interuniv. Collection: Studies on general algebra, geometry and their applications.
Chisinau: Shtiintsa, 1986, p. 104-110 (in Russian).